| Diophantine方程(a-1)x~2+f(a)=4a~n的一点注记 |
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| 引用本文： | 乐茂华.Diophantine方程(a-1)x~2+f(a)=4a~n的一点注记[J].数学学报,2011,54(1):111-114. |
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| 作者姓名： | 乐茂华 |
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| 作者单位： | 湛江师范学院数学系 湛江 524048 |
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| 基金项目： | 国家自然科学基金资助项目(10971184) |
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| 摘 要： | 设a是大于1的正整数,f(a)是a的非负整系数多项式,f(1)=2rp+4,其中r是大于1的正整数,p=2~l-1是Mersenne素数.本文讨论了方程(a-1)x~2+f(a)=4a~n的正整数解(x,n)的有限性,并且证明了:当f(a)=91a+9时,该方程仅当a=5,7和25时分别有解(x,n)=(3,3),(11,3)和(3,4).
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| 关 键 词： | 指数Diophantine方程 正整数解 有限性 |
| 收稿时间： | 2010-02-05 |
| 修稿时间： | 2010-06-10 |
A Note on the Diophantine Equation (a-1)x2+f(a)=4an |
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| Mao Hua LE.A Note on the Diophantine Equation (a-1)x2+f(a)=4an[J].Acta Mathematica Sinica,2011,54(1):111-114. |
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| Authors: | Mao Hua LE |
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| Institution: | Department of Mathematics, Zhanjiang Normal College, Zhanjiang 524048, P. R. China |
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| Abstract: | Let a be a positive integer with a>1, f(a) be a polynomial of a with nonnegative integer coefficients, and f(1)=2rp+4, where r is a positive integer with r>1, p=2l-1 is a Mersenne prime. In this paper, the finiteness of positive integer solution (x,n) of the equation
(a-1)x2+f(a)=4an
is discussed, and prove that if f(a)=91a+9, then the equation has only the positive integer solutions (x,n)=(3,3), (11,3) and (3,4) for a=5,7 and 25 respectively.
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| Keywords: | exponential diophantine equation positive integer solution finiteness |
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